Let be the length of the shorter diagonal. Then the length of the longer diagonal can be represented by .

Recall how to find the area of a kite:

Plug in the given area and solve for .

Since we are dealing with geometric shapes, the answer must be a positive value. Thus, .

The length of the shorter diagonal is units long.

Write the formula for the area of a kite.

Plug in the given diagonals.

Pull out a common factor of two in and simplify.

Use the FOIL method to simplify.

Let be the length of the shorter diagonal. Then the length of the longer diagonal can be represented by .

Recall how to find the area of a kite:

Plug in the given area and solve for .

Since we are dealing with geometric shapes, the answer must be a positive value. Thus, .

The length of the shorter diagonal is units long.

Write the formula for the area of a kite.

Plug in the given diagonals.

Pull out a common factor of two in and simplify.

Use the FOIL method to simplify.

The shape being described is a rhombus with side lengths 1. Since they are equilateral triangles connected by one side, that side becomes the lesser diagonal, so .

The greater diagonal is twice the height of the equaliteral triangles, .

The area of a rhombus is half the product of the diagonals, so:

The shape being described is a rhombus with side lengths 1. Since they are equilateral triangles connected by one side, that side becomes the lesser diagonal, so .

The greater diagonal is twice the height of the equaliteral triangles, .

The area of a rhombus is half the product of the diagonals, so:

We can find the longer diagonal by adding together the altitude of the top triangle and the altitude of the bottom triangle. To find these, use Pythagorean Theorem. We can use Pythagorean Theorem because one of the properties of a kite is that the two diagonals are perpendicular.

The top triangle has two sides of length 3 \[labeled in the picture\], and a base of 4 \[provided in the written directions\]. To figure out the altitude, split this triangle into 2 right triangles. The two legs are x \[the altitude\] and 2 \[half of the base 4\], and the hypotenuse is 3:

subtract 4 from both sides

take the square root of both sides

We will do something similar for the bottom triangle. Consider one of the right triangles. It will have a hypotenuse of 7, one leg that we don't know, x \[the altitude\], and one leg 2 \[half the shorter diagonal\]. Set up the equation using the Pythagorean Theorem:

subtract 4 from both sides

take the square root of both sides

That can be simplified by considering 45 as the product of . Since the square root of 9 is 3, we can re-write as .

Adding together the first answer of plus gives .

We can find the longer diagonal by adding together the altitude of the top triangle and the altitude of the bottom triangle. To find these, use Pythagorean Theorem. We can use Pythagorean Theorem because one of the properties of a kite is that the two diagonals are perpendicular.

The top triangle has two sides of length 3 \[labeled in the picture\], and a base of 4 \[provided in the written directions\]. To figure out the altitude, split this triangle into 2 right triangles. The two legs are x \[the altitude\] and 2 \[half of the base 4\], and the hypotenuse is 3:

subtract 4 from both sides

take the square root of both sides

We will do something similar for the bottom triangle. Consider one of the right triangles. It will have a hypotenuse of 7, one leg that we don't know, x \[the altitude\], and one leg 2 \[half the shorter diagonal\]. Set up the equation using the Pythagorean Theorem:

subtract 4 from both sides

take the square root of both sides

That can be simplified by considering 45 as the product of . Since the square root of 9 is 3, we can re-write as .

Adding together the first answer of plus gives .

Let be the length of the shorter diagonal. Then the length of the longer diagonal can be represented by .

Recall how to find the area of a kite:

Plug in the given area and solve for .

Since we are dealing with geometric shapes, the answer must be a positive value. Thus, .

The length of the shorter diagonal is units long.

Let be the length of the shorter diagonal. Then the length of the longer diagonal can be represented by .

Recall how to find the area of a kite:

Plug in the given area and solve for .

Since we are dealing with geometric shapes, the answer must be a positive value. Thus, .

The length of the shorter diagonal is units long.